Compact composition operators on the Bloch space in polydiscs

Let Uⁿ be the unit polydisc of ℂⁿ and φ=(φ₁, ⋯, φ _n ) a holomorphic self-map of Uⁿ. As the main result of the paper, it shows that the composition operator C is compact on the Bloch space β(Uⁿ) if and only if for every ε > 0, there exists a δ > 0, such that

On generalized shift bases for the wiener disc algebra

LetW(D) denote the set of functionsf(z)=Σ _n=0 ^∞ A _n Z ⁿ a _nzⁿ for which Σ_n=0 ^∞|a _n |<+∞. Given any finite set lcub;f _i (z)rcub; _i=1 ⁿ inW(D) the following are equivalent: (i) The generalized shift sequence lcub;f ₁(z)z ^kn ,f ₂(z)z ^kn+1, …,f _n (z)z ^(k+1)n−1rcub; _k=0 ^∞ is a basis forW(D) which is equivalent to the basis lcub;z ^m rcub; _m=0 ^∞ . (ii) The generalized shift sequence is complete inW(D), (iii) The function has no zero in |z|≦1, wherew=e ^2πiti /n.     

On the Asymptotic Behavior of Faber Polynomials for Domains With Piecewise Analytic Boundary

Let φ(z) be an analytic function on a punctured neighborhood of ∞, where it has a simple pole. The nth Faber polynomial F _n (z) (n=0,1,2,…) associated with φ is the polynomial part of the Laurent expansion at ∞ of [φ(z)] ⁿ . Assuming that ψ (the inverse of φ) conformally maps |w|>1 onto a domain Ω bounded by a piecewise analytic curve without cusps pointing out of Ω, and under an additional assumption concerning the “Lehman expansion” of ψ about those points of |w|=1 mapped onto corners of ∂ Ω, we obtain asymptotic formulas for F _n that yield fine results on the limiting distribution of the zeros of Faber polynomials.      

Composition operators on the Lipschitz space in polydiscs

Let Un be the unit polydisc of Cn and φ= (φ1,...,φn? a holomorphic self-map of Un. Let 0≤α< 1. This paper shows that the composition operator Cφ, is bounded on the Lipschitz space Lipa(Un) if and only if there exists M > 0 such thatfor z∈Un. Moreover Cφ is compact on Lipa(Un) if and only if Cφ is bounded on Lipa(Un) and for every ε > 0, there exists a δ > 0 such that whenever dist(φ(z),σUn) <δ     

On integral means of star-like functions

We study univalent holomorphic functions in the unit diskU = {z: |z| < 1} of the formf(z)=z+∑ _n=2 ^∞ a _n z ⁿ that satisfy the condition Re zf’(z)/f(z) > α with α∈ [0, 1) inU. Some integral means of such funcions are estimated.     

A reverse Denjoy theorem II

For α satisfying 0 < α < π, suppose that C ₁ and C ₂ are rays from the origin, C ₁: z = re ^i(π−α) and C ₂: z = re ^i(π+α), r ≥ 0, and that D = {z: | arg z − π| < α}. Let u be a nonconstant subharmonic function in the plane and define B(r, u) = sup_|z|=r u(z) and A _D (r, u) = $ \inf _{z \in \bar D_r } $ \inf _{z \in \bar D_r } u(z), where D _r = {z: z ∈ D and |z| = r}. If u(z) = (1 + o(1))B(|z|, u) as z → ∞ on C ₁ ∪ C ₂ and A _D (r, u) = o(B(r, u)) as r → ∞, then the lower order of u is at least π/(2α).     

On the growth of functions represented by Dirichlet series with complex coefficients on the real axis

We establish conditions under which, for a Dirichlet series F(z) = Σ _{n = 1} ^∞ d _n exp(λ _n z), the inequality ⋎F(x)⋎≤y(x),x≥x _o, implies the relation Σ _{n = 1} ^∞ |d _n exp(λ _n z)| ⪯ γ((1 + o(1))x) as x→+∞, where γ is a nondecreasing function on (−∞,+∞). Franko Drohobych State Pedagogic Institute, Drohobych. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1610–1616. December, 1997     

Asymptotics of Recurrence Relation Coefficients,Hankel Determinant Ratios,and Root Products Associated with Laurent Polynomials Orthogonal with Respect to Varying Exponential Weights

Let Λ^ℝ denote the linear space over ℝ spanned by z ^k , k∈ℤ. Define the real inner product 〈⋅,⋅〉 _L :Λ^ℝ×Λ^ℝ→ℝ, , N∈ℕ, where V satisfies: (i) V is real analytic on ℝ∖{0}; (ii) lim  _{| x |→∞}(V(x)/ln (x ²+1))=+∞; and (iii) lim  _{| x |→0}(V(x)/ln (x ⁻²+1))=+∞. Orthogonalisation of the (ordered) base with respect to 〈⋅,⋅〉 _L yields the even degree and odd degree orthonormal Laurent polynomials (OLPs) : φ _2n (z)=∑ _k=−n ⁿ ξ _k ⁽²ⁿ⁾ z ^k , ξ _n ⁽²ⁿ⁾>0, and φ _2n+1(z)=∑ _k=−n−1 ⁿ ξ _k ⁽²ⁿ⁺¹⁾ z ^k , ξ _−n−1⁽²ⁿ⁺¹⁾>0. Associated with the even degree and odd degree OLPs are the following two pairs of recurrence relations: z φ _2n (z)=c _2n ^♯ φ _2n−2(z)+b _2n ^♯ φ _2n−1(z)+a _2n ^♯ φ _2n (z)+b _2n+1 ^♯ φ _2n+1(z)+c _2n+2 ^♯ φ _2n+2(z) and z φ _2n+1(z)=b _2n+1 ^♯ φ _2n (z)+a _2n+1 ^♯ φ _2n+1(z)+b _2n+2 ^♯ φ _2n+2(z), where c ₀ ^♯ =b ₀ ^♯ =0, and c _2k ^♯ >0, k∈ℕ, and z ⁻¹ φ _2n+1(z)=γ _2n+1 ^♯ φ _2n−1(z)+β _2n+1 ^♯ φ _2n (z)+α _2n+1 ^♯ φ _2n+1(z)+β _2n+2 ^♯ φ _2n+2(z)+γ _2n+3 ^♯ φ _2n+3(z) and z ⁻¹ φ _2n (z)=β _2n ^♯ φ _2n−1(z)+α _2n ^♯ φ _2n (z)+β _2n+1 ^♯ φ _2n+1(z), where β ₀ ^♯ =γ ₁ ^♯ =0, β ₁ ^♯ >0, and γ _2l+1 ^♯ >0, l∈ℕ. Asymptotics in the double-scaling limit N,n→∞ such that N/n=1+o(1) of the coefficients of these two pairs of recurrence relations, Hankel determinant ratios associated with the real-valued, bi-infinite strong moment sequence , and the products of the (real) roots of the OLPs are obtained by formulating the even degree and odd degree OLP problems as matrix Riemann-Hilbert problems on ℝ, and then extracting the large-n behaviours by applying the non-linear steepest-descent method introduced in (Ann. Math. 137(2):295–368, [1993]) and further developed in (Commun. Pure Appl. Math. 48(3):277–337, [1995]) and (Int. Math. Res. Not. 6:285–299, [1997]).      

A description of Banach space-valued Orlicz hearts

Let Y be a Banach space, (Ω, Σ; μ) a probability space and φ a finite Young function. It is shown that the Y-valued Orlicz heart H _φ(μ, Y) is isometrically isomorphic to the l-completed tensor product $ H_\varphi \left( \mu \right)\tilde \otimes _l Y $ H_\varphi \left( \mu \right)\tilde \otimes _l Y of the scalar-valued Orlicz heart Hφ(μ) and Y, in the sense of Chaney and Schaefer. As an application, a characterization is given of the equality of $ \left( {H_\varphi \left( \mu \right)\tilde \otimes _l Y} \right)* $ \left( {H_\varphi \left( \mu \right)\tilde \otimes _l Y} \right)* and $ H_\varphi \left( \mu \right)*\tilde \otimes _l Y* $ H_\varphi \left( \mu \right)*\tilde \otimes _l Y* in terms of the Radon-Nikodym property on Y. Convergence of norm-bounded martingales in H _φ(μ, Y) is characterized in terms of the Radon-Nikodym property on Y. Using the associativity of the l-norm, an alternative proof is given of the known fact that for any separable Banach lattice E and any Banach space Y, E and Y have the Radon-Nikodym property if and only if $ E\tilde \otimes _l Y $ E\tilde \otimes _l Y has the Radon-Nikodym property. As a corollary, the Radon-Nikodym property in H _φ(μ, Y) is described in terms of the Radon-Nikodym property on H _φ(μ) and Y.     

Duals of holomorphic Besov spaces on the polydisks and diagonal mappings

Let U ⁿ be the unit polydisk in C ⁿ and S be the space of functions of regular variation. Let 1 ≤ p < ∞, ω = (ω ₁, ..., ω _n ), ω _j ∈ S(1 ≤ j ≤ n) and f ∈ H(U ⁿ ). The function f is said to be in holomorphic Besov space B _p (ω) if

On the order of growth of solutions of algebraic differential equations

Assume thatf is an integer transcendental solution of the differential equationP _n (z, f, f′)=P _n−1(z, f, f′, ... f ^(p)), whereP _n andP _n−1 are polynomials in all variables, the degree ofP _n with respect tof andf′ is equal ton, and the degree ofP _n−1 with respect tof, f′, ... f ^(p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E _*, whereE _* is a certain set of disks with finite sum of radii, the estimate lnf(z)=z ^1/2 (β+o(1)), β∈C, holds forz=re ^iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re ^iν)‖=o(r ^1/2),r→+∞,r>0, , where Δ is a certain set on the semiaxisr>0 with mes Δ<∞. “L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77, January, 1999.     

Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities C_n(W,P) := sup_{f ? A(W,P)}sup_{z ? W}\frac|f⁽ⁿ⁾(z)| R(f(z),P)n! (R(z,W))ⁿC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.     

On zeros of functions of given proximate formal order analytic in a half-plane

We describe sequences of zeros of functionsf≢0 that are analytic in the half-plane ℂ₊={z:Rez> and satisfy the condition

非Lipschitz条件下严格伪压缩映射不动点的逼近

Abstract. Without the Lipschitz assumption and boundedness of K in arbitrary Banach spaces,the Ishikawa iteration     

Precise rates in the law of the iterated logarithm under dependence assumptions

Let {X, X1, X2,...} be a strictly stationaryφ-mixing sequence which satisfies EX = 0,EX^2（log2{X}）^2〈∞and φ（n）=O（1/log n）^Tfor some T〉2.Let Sn=∑k=1^nXk and an=O（√n/（log2n）^γ for some γ〉1/2.We prove that limε→√2√ε^2-2∑n=3^∞1/nP（｜Sn｜≥ε√ESn^2log2n＋an）=√2.The results of Gut and Spataru （2000） are special cases of ours.     

On extremal quasiconformal extensions of conformal mappings

Letf(t, z)=z+tω(1/z) be schlicht for ⋎z⋎>1, ω(z) = Σ _n ^{= 0/∞} a _n z ⁿ ,t>0. The paper considers first-order estimates for the dilatation of extremal quasiconformal extensions off ast→0. This work was initiated during the Special Year in Complex Analysis at the Technion, and was supported in parts by the Samuel Neaman Fund, the Forschungsinstitut für Mathematik, ETH, Zürich, and the National Science Foundation.     

Some remarks on the identity between a variational integral and its relaxed functional

Summary Letf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] be measurable inx and convex inz. It is proved, by an example, that even iff verifies a condition as|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) with 1<p<q,a∈L _loc ^s (R ⁿ),s>1, the functional that isL ¹(Ω)-lower semicontinuous onW ^1,1(Ω), does not agree onW ^1,1(Ω) with its relaxed functional in the topologyL ¹(Ω) given by inf

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Riassunto Siaf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] misurabile inx e convessa inz. Si mostra con un esempio che anche sef verifica una condizione del tipo|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) con 1<p<q,a∈L _loc ^s (R ⁿ),s>1, il funzionale , che èL ¹(Ω)-semicontinuo inferiormente suW ^1,1(Ω), non coincide suW ^1,1(Ω) con il suo funzionale rilassato nella topologiaL ¹(Ω) definito da inf

     

Implicatively selector sets

Let A⊆N={0,1,2,...} and β be an n-ary Boolean function. We call A a β-implicatively selector (β-IS) set if there exists an n-ary selector general recursive function f such that (∀x₁,...,x_n)(β(χ(x₁),...,χ(x_n))=1⟹f(x₁,...,x_n)∈A), where χ is the characteristic function of A. Let F^(m), m≥1, be the family of all d _m+1 ^* -IS sets, where , F⁽⁰⁾=N, and F^(∞) is the class of all subsets in N. The basic result of the article says that the family of all β-IS sets coincides with one of F^(m), m≥0, or F^(∞), and, moreover, the inclusions F⁽⁰⁾⊂F⁽¹⁾⊂...⊂F^(∞) hold. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 145–153, March–April, 1996.     

The pointwise estimates on walsh overconvergence

Let f∈A_p. For any positive integer l, the quantity Δ_1,n−1(f:z) has been studied extensively. Here we give some quantitative estimates for and investigate some pointwise estimates of Δ _l,n−1 ^(r) (f;z). Supported by National Science Foundation of China     

Uniqueness theorem of solutions for stochastic differential equation in the plane

LetM={M _z, z ∈ R ₊ ² } be a continuous square integrable martingale andA={A _z, z ∈ R ₊ ² be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane:dX _z=α(z, X_z)dM_z+β(z, X_z)dA_z, z∈R ₊ ² , X_z=Z_z, z∈∂R ₊ ² , whereR ₊ ² =[0, +∞)×[0,+∞) and ∂R ₊ ² is its boundary,Z is a continuous stochastic process on ∂R ₊ ² . We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate (see [1, Theorem 3.2]). Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in [2]. Supported by the National Science Foundation and the Postdoctoral Science Foundation of China